Colouring of the plane, lines and circles

Tasks about colouring often occurs in mathematics, e.g. the four colour map theorem or the chromatic number of the plane. My problem begins with a case from Mathematical Olympiad: how many colours at most we can use to colour points of the plane in such a way that each line is at most two-coloured. The answer is three, but if we generalize this problem by changing two-coloured for three-coloured lines, it will be infinity many colours. We can also formulate a similar problem for circles. My work answers a more complicated question: how many colours we can use, if each line is at most m-coloured, and each circle is at most n-coloured. There are some other generalisations of the problem, e. g. changing 2D for 3D (lines for planes and circles for spheres) or changing circles for conics.